To assemble a uniformly charged sphere, assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work dW does it take to build up the radius by an amount dr? Integrate this to find the work necessary to create the entire sphere of radius R and total charge q.
W = 1/2∫ρVdτ
The Attempt at a Solution
dW = k*q*(dq/R)
This is the work needed to add a charge dq from infinity to the outer shell of the sphere
it will then increase it by dr based on the charge density added to the sphere.
so dq = 4πr^2*ρ*q*dr
making dW = (k*q*4πr^2*ρ*dr)/R
So W = k*q*4πr^2*∫ρ r^2 dr dτ
I am confused if i have set this up right and on what interval i need to integrate.